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JEE Main Vector Algebra questions

64 Mathematics questions on Vector Algebra as asked in JEE Main, newest first. 44 come with a worked solution.

  1. If the vectors a = λi + muj + 4k, b = -2i + 4j - 2k and c = 2i + 3j + k are coplanar and the projection of a on the vector b is √(54) units, then the sum of all possible values of λ + μ is equal to

    2023

  2. Let a = -i - j + k, a · b = 1 and a × b = i - j. Then a - 6b is equal to

    2023 · with solution

  3. The vector a = -i + 2j + k is rotated through a right angle, passing through the y-axis in its way and the resulting vector is b. Then the projection of 3a + √(2) b on c = 5i + 4j + 3k is:

    2023

  4. Let u = i - j - 2k, v = 2i + j - k, v · w = 2 and v × w = u + λ v. Then u · w is equal to

    2023

  5. Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that QA/AR = RB/BP = PC/CQ = 1/2. Then (Area(Δ PQR))/(Area(Δ ABC)) is equal to

    2023

  6. If the four points, whose position vectors are 3i - 4j + 2k, i + 2j - k, -2i - j + 3k and 5i - 2aj + 4k are coplanar, then a is equal to

    2023 · with solution

  7. Let α = 4i + 3j + 5k and β = i + 2j - 4k. Let β_1 be parallel to α and β_2 be perpendicular to α. If β = β_1 + β_2, then the value of 5β_2 · (i + j + k) is

    2023 · with solution

  8. Let a, b and c be three non zero vectors such that b · c = 0 and a × (b × c) = dfracb - c2. If d be a vector such that b · d = a · b, then (a × b) · (c × d) is equal to

    2023 · with solution

  9. Let a, b and c be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be a - b + c, λa - 3b + 4c, -a + 2b - 3c and 2a - 4b + 6c respectively. If overrightarrowAB, overrightarrowAC…

    2023 · with solution

  10. Let α = (λ - 2)a + b and β = (4λ - 2)a + 3b be two given vectors where vectors a and b are non-collinear. The value of λ for which vectors α and β are collinear, is

    2019 · with solution

  11. Let a = i - j, b = i + j + k and c be a vector such that a × c + b = 0 and a · c = 4, then |c|^2 is equal to

    2019 · with solution

  12. Let A(3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC. If G divides BM in the ratio 2:1, then cos(∠ GOA) (O being the origin) is equal to

    2019 · with solution

  13. The sum of the distinct real values of μ, for which the vectors, mui+j+k, i+muj+k, i+j+muk are coplanar, is

    2019 · with solution

  14. Let a = 3i + 2j + 2k and b = i + 2j - 2k be two vectors. If a vector perpendicular to both of the vectors a + b and a - b has the magnitude 12, then one such vector is

    2019

  15. Let α ∈ R and the three vectors a = αi + j + 3k, b = 2i + j - αk and c = αi - 2j + 3k. Then, the set S = \α : a, b and c are coplanar\

    2019 · with solution

  16. Let a , b and c be three unit vectors, out of which vectors b and c are non-parallel. If α and β are the angles which vector a makes with vectors b and c respectively and a × (b × c) = 1/2b, then |α - β| is equal to

    2019

  17. Let a = 2i + λ_1j + 3k, b = 4i + (3-λ_2)j + 6k and c = 3i + 6j + (λ_3 - 1)k be three vectors such that b = 2a and a is perpendicular to c. Then a possible value of (λ_1, λ_2, λ_3) is

    2019 · with solution

  18. If a unit vector a makes angles (π)/3 with i, (π)/4 with j and θ ∈ (0, π) with k, then a value of θ is

    2019

  19. Let a = i + 2j + 4k, b = i + λj + 4k and c = 2i + 4j + (λ^2 - 1)k be coplanar vectors. Then, the non-zero vector a × c is

    2019

  20. If the volume of parallelopiped formed by the vectors i + λj + k, j + λk and λi + k is minimum, then λ is equal to

    2019

  21. Let α = 3i + j and β = 2i - j + 3k. If β = β_1 - β_2, where β_1 is parallel to α and β_2 is perpendicular to α, then β_1 × β_2 is equal to

    2019

  22. Let a = i + j + √(2) k, b = b_1i + b_2j + √(2) k and c = 5i + j + √(2) k be three vectors such that the projection vector of b on a is a. If a + b is perpendicular to c, then |b| is equal to

    2019

  23. Let a = 3i + 2j + xk and b = i - j + k, for some real x. Then |a × b| = r is possible if

    2019 · with solution

  24. Let u be a vector coplanar with the vectors a = 2i + 3j - k and b = j + k. If u is perpendicular to a and u · b = 24, then |u|^2 is equal to

    2018 · with solution

  25. Let O be the origin and let PQR be an arbitrary triangle. The point S is such that OP · OQ + OR · OS = OR · OP + OQ · OS = OQ · OR + OP · OS Then the triangle PQR has S as its

    2017 · with solution

  26. Let a = 2i + j - 2k, b = i + j and c be a vector such that |c - a| = 3, |(a × b) × c| = 3 and the angle between c and a × b is 30°. Then, a · c is equal to

    2017 · with solution

  27. Let a, b and c be three unit vectors such that a × (b × c) = frac√(3)2(b + c). If b is not parallel to c, then the angle between a and b is

    2016 · with solution

  28. If a and b are vectors such that |a + b| = √(29) and a × (2i + 3j + 4k) = (2i + 3j + 4k) × b, then a possible value of (a + b) · (-7i + 2j + 3k) is

    2012 · with solution

  29. Let a = i + j + k, b = i - j + k and c = i - j - k be three vectors. A vector v in the plane of a and b, whose projection on c is frac1√(3), is given by

    2011 · with solution

  30. Two adjacent sides of a parallelogram ABCD are given by AB = 2i + 10j + 11k and AD = -i + 2j + 2k. The side AD is rotated by an acute angle α in the plane of the parallelogram so that AD becomes AD'. If AD' makes a…

    2010 · with solution

  31. Let P, Q, R and S be the points on the plane with position vectors -2i - j, 4i, 3i + 3j and -3i + 2j, respectively. The quadrilateral PQRS must be a

    2010 · with solution

  32. If a, b, c and d are the unit vectors such that (a × b) · (c × d) = 1 and a · c = 1/2, then

    2009

  33. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector a, b, c such that a · b = b · c = c · a = 1/2. Then, the volume of the parallelopiped is

    2008 · with solution

  34. Let two non-collinear unit vectors a and b form an acute angle. A point P moves, so that at any time t the position vector overrightarrowOP (where, O is the origin) is given by acos t + bsin t. When P is farthest from…

    2008

  35. The number of distinct real values of λ, for which the vectors -λ^2i + j + k, i - λ^2j + k and i + j - λ^2k are coplanar, is

    2007 · with solution

  36. Let a, b, c be unit vectors such that a + b + c = 0. Which one of the following is correct?

    2007 · with solution

  37. Let, a = i + 2j + k, b = i - j + k, c = i + j - k. A vector coplanar to a and b has a projection along c of magnitude dfrac1√(3), then the vector is

    2006 · with solution

  38. If a, b, c are three non-zero, non-coplanar vectors and b_1 = b - dfracb·a|a|^2a, b_2 = b + dfracb·a|a|^2a, c_1 = c - dfracc·a|a|^2a - dfracc·b|b|^2b, c_2 = c - dfracc·a|a|^2a - dfracc·b_1|b_1|^2b_1, c_3 = c -…

    2005

  39. The unit vector which is orthogonal to the vector 3i + 2j + 6k and is coplanar with the vectors 2i + j + k and i - j + k is

    2004 · with solution

  40. If a = (i + j + k), a · b = 1 and a × b = j - k, then b is equal to

    2003 · with solution

  41. The value of a, so that the volume of parallelopiped formed by i + aj + k, j + ak and ai + k become minimum, is

    2003 · with solution

  42. If V = 2i + j - k and W = i + 3k. If U is a unit vector, then the maximum value of the scalar triple product [U \ V \ W] is

    2002 · with solution

  43. If a and b_1 are two unit vectors such that a + 2b and 5a - 4b, are perpendicular to each other, then the angle between a and b is

    2002 · with solution

  44. If a = i - k, b = xi + j + (1-x)k and c = yi + xj + (1+x-y)k. Then, [a b c] depends on

    2001 · with solution

  45. If a, b and c are unit vectors, then |a - b|^2 + |b - c|^2 + |c - a|^2 does not exceed

    2001

  46. Let the vectors a, b, c and d be such that (a × b) × (c × d) = 0. If P_1 and P_2 are planes determined by the pairs of vectors a, b and c, d respectively, then the angle between P_1 and P_2 is

    2000 · with solution

  47. If a, b and c are unit coplanar vectors, then the scalar triple product [2a - b 2b - c 2c - a] is

    2000 · with solution

  48. If the vectors a, b and c from the sides BC, CA and AB respectively of a triangle ABC, then

    2000 · with solution

  49. Let a = 2i + j + k, b = i + 2j - k and a unit vector c be coplanar. If c is perpendicular to a, then c is equal to

    1999 · with solution

  50. Let a = 2i + j - 2k and b = i + j. If c is a vector such that a · c = |c|, |c - a| = 2√(2) and the angle between (a × b) and c is 30°, then |(a × b) × c| is equal to

    1999

  51. For three vectors u, v, w which of the following expressions is not equal to any of the remaining three?

    1998

  52. Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p × [(x - q) × p ] + q × [(x - r) × q ] + r × [(x - p) × r ] = 0 then x is given by

    1997 · with solution

  53. If a, b, c are non-coplanar unit vectors such that a × (b × c) = dfrac(b + c)√(2), then the angle between a and b is

    1995 · with solution

  54. If a, b and c are three non-coplanar vectors, then (a + b + c) · [(a + b) × (a + c)] equals

    1995

  55. Let a = i - j, b = j - k, c = k - i. If d is a unit vector such that a · d = 0 = [b c d], then d equals

    1995 · with solution

  56. Let u, v and w be vectors such that u + v + w = 0. If |u| = 3, |v| = 4 and |w| = 5, then u · v + v · w + w · u is

    1995 · with solution

  57. Let a, b, c be distinct non-negative numbers. If the vectors ai + aj + ck, i + k and ci + cj + bk lie in a plane, then c is

    1993 · with solution

  58. The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b = (0, 1, 1) is

    1987 · with solution

  59. Let a = a_1i + a_2j + a_3k, b = b_1i + b_2j + b_3k and c = c_1i + c_2j + c_3k be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b. If the angle between a and b is (π)/6, then…

    1986

  60. A vector a has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, a…

    1986

  61. The volume of the parallelopiped whose sides are given by OA = 2i - 3j, OB = i + j - k, OC = 3i - k, is

    1983 · with solution

  62. The points with position vectors 60i + 3j, 40i - 8j, ai - 52j are collinear, if

    1983 · with solution

  63. For non-zero vectors a, b, c, |(a × b) · c|= |a||b||c| holds, if and only if

    1982 · with solution

  64. Let a, b and c be three non-zero vectors such that no two of them are collinear and (a × b) × c = 1/3|b||c|a. If θ is the angle between vectors b and c, then a value of sinθ is

    with solution

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